Recall that \(\kappa _{\mathrm {ev}}\)  depends on \(k_x,\kappa \) , see (REF ). It was shown in [1]} that \(S\)  is well-defined (in particular \(g\)  is non-vanishing) as long as the boundary condition is self-adjoint and the section \(\widehat{\psi }_{\mathrm {in}}\)  and \(\widehat{\psi }_{\mathrm {out}}\)  are regular. Thus \(S(k_x,\kappa ,a)\)  is defined on
\(D_S = \Big (\mathbb {R} \times \mathbb {R}_+^* \times \mathbb {R} \Big ) \setminus \Big ( \lbrace (0,\kappa ,0) \, | \, \kappa >0\rbrace  \cup \lbrace  (0,\zeta _y,a) \, | \, a \in \mathbb {R}\rbrace \Big )\) 
